Matrices for Transformations

[Guest]

Hey,

Quick question: How do you guys remember the matrices for each transformation, like rotations?

For example, for a rotation of 90°, or -270°, the matrix is:

. . .[0 -1]
. . .[1 .0]

For 180° and -180°, it's:

. . .[-1 0]
. . .[0 -1]

For 270° or -90°, it's:

. . .[ 0 1]
. . .[-1 0]

How do you remember these matrices, along with other matrices? What's the logic to them?

 

[pka]

Just learn the general form.
\(\displaystyle \
T = \left( {\begin{array}{lr}
{\cos (\theta )} & { - \sin (\theta )} \\
{\sin (\theta )} & {\cos (\theta )} \\
\end{array}} \right)\)

 

[Guest]

Well I am self studying Algebra 2 (rather rapidly) and I still have not gotten as far as Trigonometry. I'm sure I'll be familiar with that sooner or later though!

 

[pka]

Well I am self studying Algebra 2 (rather rapidly) and I still have not gotten as far as Trigonometry. I'm sure I'll be familiar with that sooner or later though!

Sorry to say, this is an example of the perils of self study.
The whole concept of rotations is trigonometric/geometric in origin.
Your question implies that that you should know trigonometry, that is why I answered the way I did.

 

[Guest]

Well I am self studying Algebra 2 (rather rapidly) and I still have not gotten as far as Trigonometry. I'm sure I'll be familiar with that sooner or later though!

Sorry to say, this is an example of the perils of self study.
The whole concept of rotations is trigonometric/geometric in origin.
Your question implies that that you should know trigonometry, that is why I answered the way I did.

Hm? Yes, well I am using a textbook, and I've reviewed the book so I know I will be covering Trigonometry (there is a whole chapter on it) however I haven't quite gotten there yet, Matrices is chapter 4 while Trig is 10.

The Matrices I am dealing with is quite simple, but I am having trouble learning the logic between why the rotations have -1 here, or 1 there, and 0 here, or 0 there.

 

[stapel]

As was mentioned previously, the "logic" depends upon trigonometry (in order to evalute the trig functions at the given values of the angle theta). Until you have covered trigonometry, the "logic" cannot be explained.

Eliz.

 

[pka]

The Matrices I am dealing with is quite simple, but I am having trouble learning the logic between why the rotations have -1 here, or 1 there, and 0 here, or 0 there.

That is exactly what I tried to explain to you above.
Rotations are about angles.

\(\displaystyle \begin{array}{l}
\left( {\begin{array}{lr}
{\cos (90^\circ )} & { - \sin (90^\circ )} \\
{\sin (90^\circ )} & {\cos (90^\circ )} \\
\end{array}} \right) = \left( {\begin{array}{lr}
0 & { - 1} \\
1 & 0 \\
\end{array}} \right) \\
\left( {\begin{array}{lr}
{\cos (180^\circ )} & { - \sin (180^\circ )} \\
{\sin (180^\circ )} & {\cos (180^\circ )} \\
\end{array}} \right) = \left( {\begin{array}{lr}
{ - 1} & 0 \\
0 & { - 1} \\
\end{array}} \right) \\
\left( {\begin{array}{lr}
{\cos (270^\circ )} & { - \sin (270^\circ )} \\
{\sin (270^\circ )} & {\cos (270^\circ )} \\
\end{array}} \right) = \left( {\begin{array}{lr}
0 & 1 \\
{ - 1} & 0 \\
\end{array}} \right) \\
\left( {\begin{array}{lr}
{\cos (45^\circ )} & { - \sin (45^\circ )} \\
{\sin (45^\circ )} & {\cos (45^\circ )} \\
\end{array}} \right) = \left( {\begin{array}{lr}
{\sqrt 2 /2} & { - \sqrt 2 /2} \\
{\sqrt 2 /2} & {\sqrt 2 /2} \\
\end{array}} \right) \\
\end{array}\)

The last rotates 45deg.

 

[Guest]

Ah ok, I see. So for now it just means memorizing them.

Thanks!

 

[tiaa]

well my friend taught me a way which made it quite easy to learn them. it goes this way : consider all ones as l's , zeros as os and -1 as ts.

this will give the matrices a new meaning . eg:
[0 -1]
[1 0]

u can read it as otlo.

tell me if it helps.

 

[stapel]

tell me if it helps.

How does this explain the "logic" of the underlying trigonometric functions?

Please clarify. Thank you.

Eliz.